PSI - Issue 75

Marike Schwickardi et al. / Procedia Structural Integrity 75 (2025) 65–71 Schwickardi et al. / Structural Integrity Procedia (2025)

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(e.g., the 5th percentile) and inverting the excess direction. This approach was also applied to parameters such as Radius 1 and Radius 2 , where particularly small values are considered critical. 3.2 Normalization To ensure comparability across samples with different absolute scales, all values were standardized using the classical z -transformation prior to thresholding. Each sample was normalized by subtracting the sample-specific mean μ and dividing by the standard deviation σ . The same transformation was applied to the threshold, resulting in dimensionless excess values that are independent of scale and location. This allows for consistent comparison of extreme value behavior across different weld seams. , = ( − ℎ ℎ ) (2) This form retains the original exceedance above threshold , but scales it relative to the sample’s standard deviation. The result is a set of standardized excess values that are independent of scale and location , and can be meaningfully compared across different samples. 3.3 Distribution Fitting The normalized excess values from all samples were pooled and fitted against a selection of theoretical distributions that are either commonly used in extreme value statistics or serve as reference models for comparison. The goal of this step is to identify which statistical models best describe the behaviour of the extracted extremes and to provide a robust basis for interpretation and further analysis. The following statistical distributions were selected to model the excess values in the POT framework. A detailed overview of the properties and applications of these distributions can be found in Kotz and Nadarajah (2000). Generalized Pareto distribution (GPD) : The GPD is the canonical distribution for modelling exceedances in the POT framework. It is highly flexible and can represent heavy-tailed, light-tailed, and exponential-type behaviour depending on its shape parameter. It serves as a theoretical reference model in EVA. • Gumbel distribution : A special case within the family of extreme value distributions, the Gumbel distribution is often used to model the distribution of maxima in block maxima methods but can also approximate upper tails in POT analysis when the underlying data exhibits light tails. • Weibull distribution : This distribution is suitable for modelling bounded extreme values and is often used in reliability engineering and material fatigue. Its inclusion allows the detection of distributions where an upper bound might exist for weld seam features. • Exponential distribution : The exponential distribution represents the simplest case of the GPD (with shape parameter ξ=0 ). Its inclusion allows assessment of whether a memoryless, purely scale-based model can sufficiently explain the extreme values. • Log-normal distribution : While not part of classical EVA theory, the log-normal distribution is included due to its relevance in modelling skewed, multiplicative physical processes. It is often encountered in real-world measurement data, including geometry and material parameters. • Normal distribution : Included as a benchmark, the normal distribution provides a reference for symmetric, non-heavy-tailed behaviour. While not expected to perform well for extreme values, its comparison highlights the departure of the data from Gaussian assumptions. •